Every closed convex set is the set of minimizers of some $C^{\infty}$-smooth convex function

We show that for every closed convex set $C$ in a separable Banach space $X$ there is a $C^{\infty}$-smooth convex function $f:X\longrightarrow [0,\infty)$ so that $f^{-1}(0)=C$. We also deduce some interesting consequences concerning smooth approximation of closed convex sets and continuous convex functions.